Ohm's law:
voltage, current and resistance in one line

The equation in three shapes

Voltage is the push that drives charge round a circuit, current is the rate at which charge flows, and resistance is how strongly the path opposes it. Georg Ohm found that for many conductors the current is proportional to the voltage: double the push and you double the flow.

So if 2 A flows through a 6 Ω resistor, the voltage across it must be 2 × 6 = 12 V. Same circuit, same answer, every time, provided the resistor is a plain one and its temperature does not run away.

Cover the quantity you want in the familiar triangle with V on top and I and R below, and the remaining symbols tell you what to do. Milliamps must be converted to amps first: 20 mA is 0.020 A.

Power from the same numbers

The three forms of the power law are all the same equation after substituting Ohm's law. Use whichever matches the two numbers you know. For the 6 Ω resistor above, P = 2² × 6 = 24 W, which is a serious amount of heat for a small component and calls for a resistor rated to match.

Worked example: an LED resistor

A red LED needs about 2 V across it and 20 mA of current. You want to run it from a 9 V battery, so the resistor must drop the other 7 V.

Rounding up to the next standard value is the safe habit because it lowers the current slightly instead of overdriving the LED.

Resistors in series and in parallel

Real circuits seldom have one resistor. In series the values add. In parallel the reciprocals add, so the total is always smaller than the smallest branch.

• Series: R_total = R₁ + R₂. A 6 Ω and a 3 Ω in series make 9 Ω, so a 12 V supply drives 12 ÷ 9 = 1.33 A.
• Parallel: 1 ÷ R_total = 1 ÷ R₁ + 1 ÷ R₂. The same 6 Ω and 3 Ω in parallel make 2 Ω.
• Current is the same everywhere in a series loop, while voltage is the same across every parallel branch.

Voltage dividers: Ohm's law in a chain

Two resistors in series share the supply voltage in proportion to their resistance. That is a voltage divider, and it is the quickest way to get a lower voltage from a higher one for a sensor or a signal.

Put a 10 kΩ resistor and a 5 kΩ resistor in series across a 9 V supply. The current is 9 ÷ 15,000 = 0.6 mA, so the voltage across the 5 kΩ resistor is 0.0006 × 5,000 = 3.0 V, and across the 10 kΩ it is 6.0 V. Equivalently, the output is the supply multiplied by R₂ ÷ (R₁ + R₂) = 9 × 5 ÷ 15 = 3.0 V.

A divider only behaves as calculated when whatever it feeds draws little current. A load of comparable resistance in parallel with the lower resistor changes the answer, which is why dividers are poor power supplies and fine signal references.

Wire resistance and voltage drop

Wires are not perfect conductors, and Ohm's law explains the loss in long cables. A wire's resistance is R = ρ × L ÷ A, with ρ the resistivity of the metal, L the length and A the cross-section area. Copper's resistivity is about 1.7 × 10⁻⁸ Ω·m.

A 20 m run of copper wire with a cross-section of 1.5 mm² has a resistance of 1.7 × 10⁻⁸ × 20 ÷ 1.5 × 10⁻⁶ = 0.227 Ω each way, so about 0.45 Ω for the out and return path. Carrying 10 A, the cable drops 4.5 V and wastes 45 W as heat. At 230 V that is a 2% loss, but on a 12 V system it is over a third of the supply, so low-voltage installations need thicker cable.

Prefixes that keep the arithmetic honest

Most errors in circuit sums are unit errors, not conceptual ones. Circuits use prefixes constantly: milliamps, kilo-ohms, megohms, microamps. Convert everything to base units before applying V = I × R, or use matching prefixes consistently.

A handy shortcut: volts divided by kilo-ohms gives milliamps directly. A 5 V supply across a 1 kΩ resistor passes 5 mA, no conversion needed. That one habit removes most factor-of-a-thousand slips at the bench.

Where Ohm's law stops working

The law describes ohmic materials. A diode, an LED or a filament lamp does not obey a straight line between voltage and current, which is why the LED above needs a resistor to set current. A filament's resistance rises as it heats, so a lamp that reads 10 Ω cold may be 100 Ω at working temperature.

AC circuits add a further wrinkle: coils and capacitors oppose current by reactance as well as resistance, and the combined opposition is called impedance. The same form V = I × Z still holds for the magnitudes.